Intro to Business Statistics: Hypothesis Testing - Two-Sample Tests, Independent Samples t-Test for Equal and Unequal Variances Welchs t Pooled Variance t-Test

What This Is

Two-sample tests are used to compare the means of two independent samples to determine if there is a significant difference between them. A retail chain wants to know if the average daily sales of its new store location exceed $10,000, compared to its existing locations. By conducting a two-sample t-test, the chain can determine if the new location's sales are significantly higher than the existing locations.

Key Formulas & Symbols

• t = (x?1 - x?2) / sqrt((s1^2/n1) + (s2^2/n2)) where x?1 and x?2 = sample means, s1 and s2 = sample standard deviations, n1 and n2 = sample sizes.
• t = (x?1 - x?2) / sqrt((s_p^2/n1) + (s_p^2/n2)) where s_p^2 = pooled variance, x?1 and x?2 = sample means, n1 and n2 = sample sizes.
• s_p^2 = ((n1 - 1)s1^2 + (n2 - 1)s2^2) / (n1 + n2 - 2) where s1 and s2 = sample standard deviations, n1 and n2 = sample sizes.
• df = n1 + n2 - 2 where n1 and n2 = sample sizes.
• t-critical = t_(?/2, df) where-= 0.05, df = degrees of freedom.
• p-value = 2 * P(t > |t|) where t = test statistic, P = probability.
• H0: ?1 = ?2 where ?1 and ?2 = population means.
• H1: ?1-?2 where ?1 and ?2 = population means.
• ? = 0.05 where-= significance level.

Step-by-Step Procedure

1. State hypotheses: Write the null and alternative hypotheses, H0 and H1, in terms of the population means.
2. Choose test: Select the appropriate two-sample t-test (Welch's t or Pooled Variance t-Test) based on the sample variances.
3. Compute test statistic: Calculate the test statistic, t, using the selected formula.
4. Find p-value or critical value: Determine the p-value or critical value using a t-distribution table or calculator.
5. Compare to ?: Compare the p-value or critical value to the significance level, ?.
6. Conclude: Make a decision based on the comparison, rejecting H0 if the p-value is less than-or the test statistic is greater than the critical value.

Common Mistakes

• Mistake: Using the Pooled Variance t-Test when the sample variances are significantly different.
• Correction: Use Welch's t-test instead, as it is more robust to unequal variances.
• Mistake: Misinterpreting the p-value as the probability that H0 is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H0 is true.
• Mistake: Failing to check the assumptions of the t-test (independence, normality, equal variances).
• Correction: Verify that the data meet the assumptions before conducting the test.

Quick Practice Problems

1. A company wants to compare the average salaries of its male and female employees. The sample means are $50,000 and $55,000, with sample standard deviations of $5,000 and $6,000, respectively. The sample sizes are 20 and 25. What is the p-value? Answer: 0.012 (The p-value is calculated using the t-distribution table or calculator, assuming equal variances.)
2. A marketing firm wants to determine if the average response time to an advertisement is different for two different demographics. The sample means are 2 minutes and 3 minutes, with sample standard deviations of 0.5 minutes and 0.7 minutes, respectively. The sample sizes are 15 and 20. What is the test statistic? Answer: 2.45 (The test statistic is calculated using the formula for Welch's t-test.)
3. A quality control team wants to compare the average defect rate of two manufacturing processes. The sample means are 5% and 10%, with sample standard deviations of 2% and 3%, respectively. The sample sizes are 25 and 30. What is the critical value? Answer: 2.05 (The critical value is determined using the t-distribution table or calculator, assuming equal variances.)

Last-Minute Cram Sheet

• p-value is NOT the probability that H0 is true – it's the probability of observing the data (or more extreme) if H0 is true.
• Use Welch's t-test when sample variances are unequal.
• Check assumptions of t-test (independence, normality, equal variances).
• Pooled Variance t-Test assumes equal variances.
• t-critical = t_(?/2, df) where-= 0.05, df = degrees of freedom.
• p-value = 2 * P(t > |t|) where t = test statistic, P = probability.
• H0: ?1 = ?2 where ?1 and ?2 = population means.
• H1: ?1-?2 where ?1 and ?2 = population means.
• ? = 0.05 where-= significance level.
• df = n1 + n2 - 2 where n1 and n2 = sample sizes.
• s_p^2 = ((n1 - 1)s1^2 + (n2 - 1)s2^2) / (n1 + n2 - 2) where s1 and s2 = sample standard deviations, n1 and n2 = sample sizes.